To compute this virtual degree, we note that by [1]}, there exists a log algebraic stack \(\mathfrak {M}^{\prime }(\tilde{\mathcal {X}} \rightarrow \mathcal {X})\) , and a strict étale morphism \(\mathfrak {M}^{\prime }(\tilde{\mathcal {X}} \rightarrow \mathcal {X}) \rightarrow \mathfrak {M}(\tilde{\mathcal {X}})\) , such that there exists the following cartesian diagram:
\(\begin{tikzcd}\mathcal {M}(\tilde{X},\beta ^{\prime }){r} {d} & \mathfrak {M}^{\prime }(\tilde{\mathcal {X}} \rightarrow \mathcal {X}) {d} \\\mathcal {M}(X,\beta ) {r} & \mathfrak {M}(\mathcal {X})\end{tikzcd}\) 
